
Unfortunately, the later estimate is far from sharp. In [1]}, the following bound is given.
\(\frac{1}{2 \gamma _{\rm {ps}}} \le \tau _{\rm {mix}}\le \frac{1}{\gamma _{\rm {ps}}}\bigg (\ln \frac{1}{\pi _*}+2 \ln 2+1 \bigg ).\) 
